Finite Group Databasebeta

← #412402

Subgroups — #412402

Order 512.

1620 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#467C211yesmaximal, characteristic
2C2×#2035C212yesmaximal
3C2×#1969C211yesmaximal, characteristic, complemented
4C2×#1969C211yesmaximal, characteristic, complemented
5C2×#1846C211yesmaximal, characteristic, complemented
6C2×#1808C211yesmaximal, characteristic
7C2×C8×C22⋊C4C211yesmaximal, characteristic
8#15236C212yesmaximal
9#11122C212yesmaximal, complemented
10#11052C212yesmaximal, complemented
11#10862C211yesmaximal, characteristic, complemented
12C23×Q16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C2×#467—24no—
16C2×#467—24no—
17C2×#467—24no—
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#106C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×#103C2212yes—
25C22×C22⋊C8C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#382C2212yes—
31C2×#380C2212yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33#2035—28no—
34#1969—24no—
35#1969—24no—
36#1846—24no—
37#1808—24no—
38C8×C22⋊C4—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1442—24no—
44#1440—24no—
45C4⋊#95C2212yes—
46C23×Q8C2312yes—
47C24×C4C2311yescharacteristic
48C22×Q16—22no—
49C22×Q16—22no—
50C22×Q16—22no—
51C22×Q16—22no—
52C22×Q16—22no—
53C22×Q16—22no—
54C22×Q16D414yes—
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8D414yes—
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8—42no—
71C2×C22⋊Q8—42no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4D412yes—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81#467—232no—
82C2×#107—22no—
83C2×#107—22no—
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#107D414yes—
91C2×#106—24no—
92C2×#106—24no—
93C2×#106—42no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—42no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8D414yes—
106C2×C4×C8—22no—
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110C2×C4×C8—22no—
111C2×C4×C8D414yes—
112#391—24no—
113#391—24no—
114#391—24no—
115#391—42no—
116#382—24no—
117#382—42no—
118#380—24no—
119#380—24no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C2×#95—24no—
123C2×#95—24no—
124C2×#95—24no—
125C2×#95—24no—
126C2×#95—22no—
127C2×#95D412yes—
128#340—24no—
129#340—24no—
130#336—22no—
131#336—22no—
132C25C2×D411yescharacteristic
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8C2×D414yes—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2411yescharacteristic
169C2×Q16—28no—
170C2×Q16—28no—
171C2×Q16—28no—
172C2×Q16—44no—
173C2×Q16—44no—
174C2×Q16—44no—
175C2×Q16—44no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—28no—
193C22×C8—28no—
194C22×C8—28no—
195C22×C8—44no—
196C22×C8—44no—
197C22×C8—44no—
198C22×C8—44no—
199C22×C8C2×D412yes—
200C22×C8C2×D412yes—
201C22×C8C2×D412yes—
202C22×C8C2×D412yes—
203C22×C8C2×D412yes—
204C22×C8C2×D412yes—
205C22×C8C2×D412yes—
206C22×C8C2×D412yes—
207C22×C8#4112yes—
208C22×C8#4112yes—
209C22×C8#4112yes—
210C22×C8#4112yes—
211C22⋊Q8—216no—
212C22⋊Q8—48no—
213C22⋊Q8—48no—
214C22⋊Q8—48no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4C2×D414yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—48no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42C2×D412yes—
255C2×C42#4112yes—
256#107—28no—
257#107—44no—
258#107—44no—
259#106—48no—
260#103—216no—
261#103—48no—
262#103—48no—
263C22⋊C8—28no—
264C4×C8—28no—
265C4×C8—44no—
266#95—24no—
267#95—48no—
268#95—48no—
269#95—48no—
270#95—44no—
271#95—48no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24#12712yes—
282C24#13514yes—
283C24C22×D411yescharacteristic
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4#12114yes—
381C22×C4#12712yes—
382C22×C4#13514yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4H(5,𝔽2)11yescharacteristic
394C22×C4H(5,𝔽2)11yescharacteristic
395C22×C42−1+411yescharacteristic
396C22×C42−1+411yescharacteristic
397Q16—416no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—416no—
419C2×C8#12014yes—
420C2×C8#12114yes—
421C2×C8#12114yes—
422C2×C8#12114yes—
423C2×C8#12114yes—
424C2×C8#12314yes—
425C2×C8#12314yes—
426C2×C8#12414yes—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—416no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C22⋊C4—216no—
441C22⋊C4—416no—
442C22⋊C4—88no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23—41no—
472C23—42no—
473C23#49514yes—
474C23C2×#12111yescharacteristic
475C23C2×#12711yescharacteristic
476C23#53211yescharacteristic
477C23D4211yescharacteristic
478C23D4211yescharacteristic
479C23#54711yescharacteristic
480C23#54711yescharacteristic
481C23C2×#13512yes—
482C23C2×#13512yes—
483C23#57812yes—
484C23#57812yes—
485Q8—44no—
486Q8—44no—
487Q8—44no—
488Q8—44no—
489Q8—82no—
490Q8—82no—
491Q8—82no—
492Q8—82no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—48no—
526C2×C4—44no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—84no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—84no—
540C2×C4—82no—
541C2×C4—84no—
542C2×C4—84no—
543C2×C4—84no—
544C2×C4C2×#12011yescharacteristic
545C2×C4C2×#12111yescharacteristic
546C2×C4C2×#12111yescharacteristic
547C2×C4C2×#12111yescharacteristic
548C2×C4C2×#12111yescharacteristic
549C2×C4C2×#12311yescharacteristic
550C2×C4C2×#12311yescharacteristic
551C2×C4C2×#12411yescharacteristic
552C2×C4#53211yescharacteristic
553C2×C4#53211yescharacteristic
554C2×C4#53511yescharacteristic
555C2×C4#53611yescharacteristic
556C2×C4#53711yescharacteristic
557C2×C4#53711yescharacteristic
558C2×C4#54211yescharacteristic
559C2×C4#54211yescharacteristic
560C2×C4D4211yescharacteristic
561C2×C4D4211yescharacteristic
562C2×C4#54711yescharacteristic
563C2×C4#54711yescharacteristic
564C2×C4#54711yescharacteristic
565C2×C4#54711yescharacteristic
566C2×C4#54811yescharacteristic
567C2×C4#54811yescharacteristic
568C2×C4#55311yescharacteristic
569C2×C4#55311yescharacteristic
570C2×C4#55511yescharacteristic
571C2×C4#55511yescharacteristic
572C8—24no—
573C8—42no—
574C8—42no—
575C8#39418yes—
576C22—24no—
577C22—24no—
578C22—24no—
579C22—42no—
580C22—42no—
581C22—44no—
582C22—44no—
583C22—42no—
584C22—42no—
585C22—44no—
586C22—44no—
587C22#241611yescharacteristic
588C22#242211yescharacteristic
589C22#243311yescharacteristic
590C22#249111yescharacteristic
591C22#250111yescharacteristic
592C22#253711yescharacteristic
593C22#285611yescharacteristic
594C22#305012yes—
595C22#305912yes—
596C22C2×#49511yescharacteristic
597C22#314711yescharacteristic
598C22#315611yescharacteristic
599C22#315611yescharacteristic
600C22#316812yes—
601C22#317712yes—
602C22#328312yes—
603C22#328512yes—
604C22#328512yes—
605C22#328512yes—
606C22#333612yes—
607C22#333612yes—
608C22#333612yes—
609C22#334212yes—
610C4—42no—
611C4—44no—
612C4—44no—
613C4—84no—
614C4—84no—
615C4—162no—
616C4—162no—
617C4C2×#39411yescharacteristic
618C4#241611yescharacteristic
619C4#242211yescharacteristic
620C4#243211yescharacteristic
621C4#249111yescharacteristic
622C4#250011yescharacteristic
623C4#253711yescharacteristic
624C4#262011yescharacteristic
625C2—44no—
626C2#1513011yescharacteristic
627C2#2477912yes—
628C2#2503312yes—
629C2#2577811yescharacteristic
630C2#2578211yescharacteristic
631C2#2778912yes—
632C2#2792711yescharacteristic
633C2#2793111yescharacteristic
634C2#2822612yes—
635C2#3317311yescharacteristic
636C2#3318011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#1969C2semidirect
4. C2×#1969C2semidirect
5. C2×#1846C2semidirect
9. #11122C2semidirect
10. #11052C2semidirect
11. #10862C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#4672
C2×#20352
C2×#19692
C2×#18462
C2×#18082
C2×C8×C22⋊C42
#152362
#111222
#110522
#108622